Classification of Singularities of Smooth Functions with a Finite Cyclic Symmetry Group

Abstract

In the paper, we study the singularities of smooth functions of two variables that are invariant under the action of a finite group \(G\) acting by rotations. A classification is obtained for critical points arising in typical parametric families of \(G\)-invariant smooth functions with at most two parameters, when \(|G|\ne4\). A criterion is obtained for the reducibility of a smooth \(G\)-invariant function to a normal form (by means of a \(G\)-equivariant change of variables) when the Taylor polynomial of degree \(|G|\) of the function is not a polynomial in \(x^2+y^2\) and the Milnor \(G\)-multiplicity (the \(G\)-codimension, respectively) of the singularity is less than \(|G|\) (than \(|G|/2\), respectively). A criterion is obtained for the reducibility of a smooth parametric family of \(G\)-invariant functions to a normal form near the critical point of the type in question. The criteria are given in terms of partial derivatives of the function at the critical point.